Solve the following problem show all of the steps 4 y y yx0

Solve the following problem. show all of the steps.

4. y\' = |y|, y(x0) = y0 Apply Lindelof?s theorem to find the largest interval(s) for x0 where the initial value problem has a unique solution.

Solution

y\' =-y if y <0 and

y\' =y if y>0

Let us find solutions for both parts

First part gives ln y =-x or y = e^-x +C\'

and II part gives y = e^x+C

But as y being a power of e is always positive and can never be below 0.

y = e^x+C

Substitute x0 and y0 to get C

C = y0-e^x0

^{------------------------------------------------------}

y =  e^x+ y0-e^x0